The limiting spectral distribution of large random permutation matrices

We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let X = ( x₁,…, xₙ) ∈ C m × n be an m × n data matrix after self-normalization ( n samples and m features), where xⱼ = (x₁ⱼ*,…, xₘⱼ* )* . Specifically, we generate a permutation matrix X_π by permuting the entries of xⱼ (j=1,…,n) and demonstrate that the empirical spectral distribution of Bₙ = (m/n)U ₙ X _π X _π* U ₙ* weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where U ₙ is a sequence of p × m non-random complex matrices. The conditions we require are p/n → c >0 and m/n → γ > 0 .